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The problem involves using trigonometry to determine the vertical distance between the goldfinch and the base of the pole, given the angle of depression and the horizontal distance from the pole.

### Information given:
- Horizontal distance from the goldfinch to the base of the pole: 13 meters
- Angle of depression: 32°
- We need to find the **vertical distance**, which we'll call \( h \).

### Approach:
Since the angle of depression is measured from the horizontal, it forms a right-angled triangle. In this triangle:
- The horizontal distance from the goldfinch to the base of the pole is adjacent to the angle.
- The vertical distance from the goldfinch to the base of the pole is the opposite side.
- We use the tangent function because tangent relates the opposite and adjacent sides:
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{13}
  \]
  where \( \theta = 32^\circ \).

Rearranging the formula to solve for \( h \):
\[
h = 13 \times \tan(32^\circ)
\]

Now, let's compute the value of \( h \).The vertical distance between the goldfinch and the base of the pole is approximately **8.12 meters** (to 2 decimal places).

Would you like more details or have any other questions? Here are some related questions:

1. How would the result change if the angle of depression were increased?
2. How do you calculate the angle of depression if the vertical distance is known?
3. Can you apply this method to find the height of an object using an observer on the ground?
4. What trigonometric function would you use if you knew the hypotenuse instead of the horizontal distance?
5. How would you solve this if the distance were given in feet rather than meters?

**Tip:** In right-angle trigonometry, always ensure you're using the correct ratio (sine, cosine, tangent) based on the given sides of the triangle.

A goldfinch is flying 13 meters away from a bird box that is on top of a pole. The angle of depression from the goldfinch to the base of the pole is 32°. What is the distance between the goldfinch and the base of the pole? Give your answer to 2 d.p.

Learn how to calculate the distance between a goldfinch and the base of a pole using trigonometry. This problem involves the angle of depression (32°) and a horizontal distance of 13 meters. Suitable for students in Grades 8-10.

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